Jackfull gets a bonus point for resourcefulness.                

Did everyone else finish their homework? Any questions? Some of you got stuck? Does this help?

Some of you probably remember the Pythagorean Theorem: the sum of the squares of the legs on a right triangle equals the square of the hypotenuse. Which means:

(Apologies for the typographic awkwardness here. The superscript formatting feature of the editor doesn't seem to do what it is intended to do so I'm using a caret (^) as a prefix for an exponent. So "x squared" is written "x^2")

(r-y)^2 + (x/2)^2 = r^2

It would be nice if we could simply take the square root of both sides to determine what the radius r is, but in order to solve for r all of the rs have to be together and there’s another one lurking on the left side of the equation. So, let’s expand the equation by squaring the binomial (r-y) and the fraction (x/2):

r^2-2ry+y^2 + x^2/4 = r^2

Look at that, now we have r^2 on both sides of the equation. As you recall we can perform any operation we like to one side of an equation so long as we do the same thing to the other. Let’s subtract r^2 from both sides:

-2ry+y^2 + x^2/4 = 0

Now let’s add 2ry to both sides:

y2 + x2/4 = 2ry

We can isolate the r now by dividing both sides by 2y:

(y^2 + x^2/4)/2y = r

There we have it, a formula for the radius of an arc given a chord length and the distance to the arc along the perpendicular bisector of the chord.

Only two of my students managed to work that out on their own, but almost all of them fully understood the application of the Pythagorean Theorem to the situation, and most of them were able to follow the derivation of the formula.

Enough of that.

Once the track centerlines were drawn on the butcher paper, students traced them with the point of a compass to draw offset lines on either side representing the cut lines. For economy I determined locations for dividing diverging track pieces so we could get all of the roadbed from a single 4x8 sheet of roadbed material.

Although I cheated by working it out in advance to be sure it could be done, I left it as a puzzle for the kids to actually lay out the patterns on the 4x8 material (Homosote).

Some of the kids had done some sewing before and were familiar with the process of using paper patterns to lay out and cut complex shapes from sheet material. They were very helpful to the students who were doing this for the first time.

Once they had it all laid out, it was quick work for a team to trace around the patterns.

So far we’d limited our use of power tools to a cordless drill for pilot holes. I considered the benefits and limitations of bringing in manual keyhole saws for cutting out the roadbed, but we were close to the end of the school year and I was hoping to get far enough along to run a train before we packed everything up for the summer. So I brought in a jigsaw.

Jeff Allen